Certainly an easy question for you (specialists on this forum) but absolutely not clear for me. The ADM Approach (1 + 3), as it can be seen in Misner Thorne and Wheeler or some other reference, is based on a kind of extension of the Pythagorras Theorem to a Minkowki space. Ok. Its signature is (- +++); convention, I suppose; ok. And then you can see in some equations resulting of this approach a (3-3) matrix supposed to represent the local 3D metric tensor as if it could have any value and not obligatory the unit matrix I(3) which is the spatial part of the (4-4) one associated to a Lorentz metric. Why? Does it come from the manner to "cut" the slice of time? I am sure something is not ok in my head concerning this point. Thanks for help.